This can be extended to the case \(\widehat{G}^{(2)}(\rho ) =\infty \)  by looking at the first two truncated moments, as in e.g. [1]}. We reproduce the argument for the reader's convenience. Let \(L_{1}\)  be the slowly varying function defined by \(L_{1}(x) = \mathbb {E}[\widehat{X}_{1}^{2} \mathbb {1}_{\lbrace |\widehat{X}_{1}| \leqslant x\rbrace }]\)  for every \(x > 0\) .
As in e.g. [2]} we have \((L_{1}(x)-m) \sim 2 L(x)\)  as \(x \rightarrow \infty \) . Further, by [3]}, the ratio \(x^2 \mathbb {P}(\widehat{X}_{1} \geqslant x) / L_{1}(x)\)  converges to \((2-\alpha )/2 = 0\)  as \(x \rightarrow \infty \) .
Then for every \(\varepsilon > 0\) , since \(L_1\)  is slowly varying, then \(L_1(\varepsilon r_n) \sim L_1(r_n)\)  and so
\(\mathbb {P}\mathopen {}\mathclose {\left(r_n^{-1} \max _{1 \leqslant i \leqslant n} \widehat{X}_i \geqslant \varepsilon \right)}\leqslant n \mathbb {P}\mathopen {}\mathclose {\left(\overline{X}_{1} \geqslant \varepsilon r_n\right)}= o\mathopen {}\mathclose {\left(n (\varepsilon r_n)^{-2} L_{1}(\varepsilon r_n)\right)}= o\mathopen {}\mathclose {\left(n r_n^{-2} L_{1}(r_n)\right)}= o(1),\) 
